The Verification Venue · a promise the dictionary can't keep

The Dictionary That Eats Itself

Every word in the dictionary is defined using other words in the same dictionary. So the advice — just look up the words you don't know — is a promise it structurally cannot keep. Follow it and the lookups never bottom out. They fall into a knot of words that only define each other, and there is no way down to the ground.

You already half-know this. But "dictionaries are circular, so what?" is a shrug, not a fact. Below, the shrug is turned into something you can measure. Pick a word. Look it up. Then look up the words in that definition. Keep your finger on the trail and watch where it goes — on a real, public-domain dictionary, with the whole computation happening in front of you.

Every underlined word in that definition is itself an entry in the same book. Click one and you're reading a new definition — built, again, from words that are entries. Try to reach a word that is defined in terms of nothing you have to look up. You can't. Within a handful of clicks the trail turns red: you're back at a word you already passed. That is not bad luck. It is the shape of the whole dictionary.

What you fell into has a name

Represent the dictionary as a graph: draw an arrow from a word to each content-word used in its definition — word → sign, word → sound, word → idea — the "to understand this, look that up" direction.^① Now ask which words you can reach from which. Two words are in the same strongly-connected component when there's a chain of lookups from each to the other and back. In 1972 Robert Tarjan gave a way to find every such component in one linear pass. Run it on this dictionary and almost the entire thing collapses into a single component.

The site does exactly that below — no numbers are typed in; Tarjan's algorithm runs on the shipped lexicon when the page loads, and prints what it finds.

The reachable set stops growing at 739 words — and every single lookup you could still make from inside it lands on a word already in the set. There is no exit. Here is the whole structure, recomputed live from the shipped file:

One component of 724 words; the next-biggest is 3. That is the shape the paper by Philippe Vincent-Lamarre, Stevan Harnad and colleagues found in 2016 in learner dictionaries and WordNet, and it is the shape here too: one giant knot, and everything else a scatter of tiny fragments hanging off it. Of every lookup you can make starting from a word inside the Core, 98.6% lands you on another word inside the Core. It is a near-perfect trap.

“So what?” has a precise answer

The shrug — of course a dictionary is circular — assumes the circularity is a shapeless mess. It isn't. The 2016 result carves it into three nested objects, each of which you can compute:

The Kernel. Start deleting words that no definition ever uses — words that are only defined, never defining. Deleting them frees up others, so repeat until you can't. What survives is the Kernel: 755 words here, the part of the dictionary that actually does the defining. Everything you looked up on the way in was, ultimately, built out of these.

The Core. Inside the Kernel sits the strongly-connected knot you already met — the 724-word component where every word has a path to and from every other. This is the part that can never be broken into "define A first, then B, then C." There is no first. Whatever order you try, some definition points backward.

The Minimum Grounding Set. Here is the sharp question. Suppose someone simply gave you the meanings of a few words — grounded them from outside the book. What is the smallest set you'd need, so that every other word could then be defined without any circle left? That set is a minimum feedback vertex set: the fewest words whose removal makes the whole tangle acyclic. And this is where the tidy story fractures — twice.

First: finding the true minimum on the full Core is NP-hard. Nobody can hand you the minimum grounding set of English by running an algorithm to completion. Second, and stranger: even where you can compute it, it is not unique. There is no such thing as the irreducible core of the language — there are many equally-small sets, each of which would do. Watch both happen on a piece of the real Core small enough to solve exactly:

Three different pairs of words — {cause, act}, {cause, power}, {act, make} — each of which, if you already knew it, would let you define the other four without any circle. No pair is the foundation. "The irreducible core of English" is not a well-defined thing; it is a choice among many equally-good ones. That is the twist the shrug never saw coming: the circularity is real, it is measurable, and pinning down its "bottom" is both computationally hard and genuinely ambiguous.

Where the graph stops and the argument begins

Everything up to here is fact. The Core exists; you can compute it; it is one enormous strongly-connected component. But there's a famous next step, and it is not a theorem — it is an argument, and it is still contested.

In 1990 Stevan Harnad posed the symbol grounding problem: a symbol system whose symbols are only ever defined by other symbols is like trying to learn Chinese from a Chinese-only dictionary — you can shuffle forever and never break out to what any word means. Meaning, he argued, is parasitic: the words in your head feel meaningful only because you ground some of them in sensorimotor experience — in seeing red, in lifting a weight, in the world. Twenty-six years later, Harnad was a co-author of the paper that measured the trap he'd described: the Core is what a purely word-defined system can never escape on its own.

One more distinction worth keeping straight. Publishers already act as if a small grounding set exists: the Longman learner's dictionary writes every definition using a fixed list of about 2,000 words — its "defining vocabulary" — and prints anything outside it in small capitals. That list is human-chosen and pedagogical; the Kernel and the Minimum Grounding Set are computed. They are different objects, arrived at differently, and they are different sizes — but both are the same instinct, from opposite ends: that under all the words, a much smaller set is quietly doing the work.